Classical Extended Kalman Filter Research Articles

Inverse optimal controller based on extended Kalman filter for discrete‐time nonlinear systems

SummaryIn this study, we present an inverse optimal control approach based on extended Kalman filter (EKF) algorithm to solve the optimal control problem of discrete‐time affine nonlinear systems. The main aim of inverse optimal control is to circumvent the tedious task of solving the Hamilton‐Jacobi‐Bellman equation that results from the classical solution of a nonlinear optimal control problem. Here, the inverse optimal controller is based on defining an appropriate quadratic control Lyapunov function (CLF) where the parameters of this candidate CLF were estimated by adopting the EKF equations. The root mean square error of the system states is used as the observed error in the case of classical EKF algorithm application, whereas, here, the EKF tries to eliminate the same root mean square error defined over the parameters by generating a CLF matrix with appropriate elements. The performance and the applicability of the proposed scheme is illustrated through both simulations performed on a nonlinear system model and a real‐time laboratory experiment. Simulation study demonstrate the effectiveness of the proposed method in comparison with 2 other inverse control approaches. Finally, the proposed controller is implemented on a professional control board to stabilize a DC‐DC boost converter and minimize a meaningful cost function. The experimental results show the applicability and effectiveness of the proposed EKF‐based inverse optimal control even in real‐time control systems with a very short time constant.

State of Charge Estimation for Lithium-Ion Battery Based on Nonlinear Observer: An H∞ Method

This work is focused on the state of charge (SOC) estimation of a lithium-ion battery based on a nonlinear observer. First, the second-order resistor-capacitor (RC) model of the battery pack is introduced by utilizing the physical behavior of the battery. Then, for the nonlinear function of the RC model, a one-sided Lipschitz condition is proposed to ensure that the nonlinear function can play a positive role in the observer design. After that, a nonlinear observer design criterion is presented based on the H ∞ method, which is formulated as linear matrix inequalities (LMIs). Compared with existing nonlinear observer-based SOC estimation methods, the proposed observer design criterion does not depend on any estimates of the unknown variables. Consequently, the convergence of the proposed nonlinear observer is guaranteed for any operating conditions. Finally, both the static and dynamic experimental cases are given to show the efficiency of the proposed nonlinear observer by comparing with the classic extended Kalman filter (EKF).

Cooperative Source Node Tracking in Non-Line-of-Sight Environments

The accuracy of localization is highly degraded in indoor and harsh environments where source nodes either do not have connections with a sufficient number of anchor nodes due to strong attenuation or have very poor range estimates due to NLOS propagation. Cooperative localization is a technique in which the source nodes communicate not only with the anchor nodes, but also with each other. Hence, the source nodes can collect several additional measurements which significantly improve the localization performance. Although many studies have examined NLOS-degraded localization of a static node in noncooperative networks, and many others have examined the impact of cooperation for static localization, there is no work which considers cooperative tracking of mobile nodes. To address this open problem, in this work, we examine cooperative tracking, particularly in NLOS environments. More specifically, we develop a novel sensor tracking algorithm based on semidefinite programming (SDP) which has the ability to mitigate NLOS propagation. Our simulations show that the new SDP-based tracking algorithm outperforms the classic extended Kalman filter as well as the other recently proposed algorithms for noncooperative tracking in NLOS environments. We also show that the algorithm can be extended to cooperative networks, and that a substantial performance benefit is realized by cooperation.

On fault detection and exclusion in snapshot and recursive positioning algorithms for maritime applications

Introduction Resilient provision of Position, Navigation and Timing (PNT) data can be considered as a key element of the e-Navigation strategy developed by the International Maritime Organization (IMO). An indication of reliability has been identified as a high level user need with respect to PNT data to be supplied by electronic navigation means. The paper concentrates on the Fault Detection and Exclusion (FDE) component of the Integrity Monitoring (IM) for navigation systems based both on pure GNSS (Global Navigation Satellite Systems) as well as on hybrid GNSS/inertial measurements. Here a PNT-data processing Unit will be responsible for both the integration of data provided by all available on-board sensors as well as for the IM functionality. The IM mechanism can be seen as an instantaneous decision criterion for using or not using the system and, therefore, constitutes a key component within a process of provision of reliable navigational data in future navigation systems. Methods The performance of the FDE functionality is demonstrated for a pure GNSS-based snapshot weighted iterative least-square (WLS) solution, a GNSS-based Extended Kalman Filter (EKF) as well as for a classical error-state tightly-coupled EKF for the hybrid GNSS/inertial system. Pure GNSS approaches are evaluated by combining true measurement data collected in port operation scenario with artificially induced measurement faults, while for the hybrid navigation system the measurement data in an open sea scenario with native GNSS measurement faults have been employed. Results First, the performance of the proposed FDE schemes in terms of the horizontal error is evaluated for both weighted and unweighted approaches in GNSS-based snapshot and KF-based schemes. Here, mainly due to availability of the process model, the KF approaches have demonstrated smaller sensitivity to the injected GNSS faults, while the methods with CNo weighting schemes have resulted in reduced spread of the obtained position solutions. The statistical evaluation of the proposed FDE schemes have been performed for pure GNSS schemes by considering the fault detection rate as a function of the amplitude for the randomly injected GNSS faults. Although the KF-based approaches have clearly outperformed the memoryless schemes, lower detection rates for weighted schemes could be clearly seen due to inability of the FDE to detect faults of fixed amplitude for satellites with lower CNo values. Moreover, the evaluation of the FDE schemes in terms of maximum horizontal position error have indicated bounded response of the FDE schemes when compared to that of non-FDE methods. Finally, the superiority of the FDE-enabled tightly-coupled GNSS/inertial EKF over the non-FDE solution have been demonstrated using a scenario with native GNSS faults. Conclusions The work had successfully demonstrated an applicability of the developed FDE schemes in snapshot and RBE-based algorithms for maritime applications using both non-inertial GNSS-based positioning and a hybrid IMU/GNSS EKF-based approach. The proposed methods form a solid foundation for construction of a more reliable and robust PNT-Unit, where state-of-the-art hybrid navigation algorithms are augmented with integrity monitoring functionality to ensure the system performance in the presence of GNSS faults. The FDE mechanism provides consistent improvements in terms of the horizontal accuracy both in LS and KF-based methods. Although only port operation case and one example of a true GNSS fault in open sea was considered, the presented results are believed to be general enough and the scheme could be adopted for other applications in future.

Moving-window extended Kalman filter for structural damage detection with unknown process and measurement noises

The extended Kalman filter (EKF), as a popular tool for optimally estimating system state from noisy measurement, has been used successfully in various areas over the past several decades. However, classical EKF has several limitations when applied to structural system identification; thus, researchers have proposed a number of variations for this method. The current study focuses on using EKF for real-time system identification and damage detection in civil structures. An improved EKF approach, called moving-window EKF (MWEKF), is proposed in this paper after a discussion on the problems associated with the application of classical EKF in time-variant systems. The proposed approach uses the moving-window technique to estimate several statistical properties. MWEKF is more robust and adaptive in structural damage detection compared with classical EKF because of the following reasons: (1) it is insensitive to the selection of the initial state vector; (2) it exhibits more accurate system parameter identification; and (3) it is immune to the inaccurate assumption of noise levels because measurement and process noise levels are estimated in this approach. The salient features of MWEKF are illustrated through numerical simulations of time-variant structural systems and an experiment on a three-story steel shear building model. Results demonstrate that MWEKF is a robust and effective tool for system identification and damage detection in civil structures.

Nonlinear ego-motion estimation from optical flow for online control of a quadrotor UAV

For the control of unmanned aerial vehicles (UAVs) in GPS-denied environments, cameras have been widely exploited as the main sensory modality for addressing the UAV state estimation problem. However, the use of visual information for ego-motion estimation presents several theoretical and practical difficulties, such as data association, occlusions, and lack of direct metric information when exploiting monocular cameras. In this paper, we address these issues by considering a quadrotor UAV equipped with an onboard monocular camera and an inertial measurement unit (IMU). First, we propose a robust ego-motion estimation algorithm for recovering the UAV scaled linear velocity and angular velocity from optical flow by exploiting the so-called continuous homography constraint in the presence of planar scenes. Then, we address the problem of retrieving the (unknown) metric scale by fusing the visual information with measurements from the onboard IMU. To this end, two different estimation strategies are proposed and critically compared: a first exploiting the classical extended Kalman filter (EKF) formulation, and a second one based on a novel nonlinear estimation framework. The main advantage of the latter scheme lies in the possibility of imposing a desired transient response to the estimation error when the camera moves with a constant acceleration norm with respect to the observed plane. We indeed show that, when compared against the EKF on the same trajectory and sensory data, the nonlinear scheme yields considerably superior performance in terms of convergence rate and predictability of the estimation. The paper is then concluded by an extensive experimental validation, including an onboard closed-loop control of a real quadrotor UAV meant to demonstrate the robustness of our approach in real-world conditions.

Adaptive fuzzy neuro-observer applied to low cost INS/GPS

A combined MEMS Inertial Navigation System (INS) with GPS is used to provide position and velocity data of land vehicles. Data fusion of INS and GPS measurements are commonly achieved through a conventional Extended Kalman filter (EKF). Considering the required accurate model of system together with perfect knowledge of predefined error models, the performance of the EKF is decreased due to unmodeled nonlinearities and unknown bias uncertainties of MEMS inertial sensors. Universal knowledge based approximators comprising of neural networks and fuzzy logic methods are capable of approximating the nonlinearities and the uncertainties of practical systems. First, in this paper, a new fuzzy neural network (FNN) function approximator is used to model unknown nonlinear systems. Second, the process of design and real-time implementation of an adaptive fuzzy neuro-observer (AFNO) in integrated low-cost INS/GPS positioning systems is proposed. To assess the long time performance of the proposed AFNO method, wide range tests of a real INS/GPS with a car vehicle have been performed. The unbiased estimation results of the AFNO show the superiority of the proposed method compared with the classic EKF and the adaptive neuro-observer (ANO) including a pure artificial neural network (ANN) function approximator.

Attitude Estimation in SO(3): A Comparative UAV Case Study

This paper introduces a novel algorithm to obtain attitude estimations from low cost inertial measurement units including 3-axis accelerometer, 3-axis gyroscope and 3-axis magnetometer. This nonlinear attitude estimator is derived from Lyapunov's theory and formulated in the special orthogonal group SO(3). The impact of the gyroscope bias is also assessed and an online estimator provided. The performance of the proposed estimator is validated and compared to current commonly used methods, namely the classical extended Kalman filter and two other nonlinear estimators in SO(3). Realistic simulations consider a quadcopter unmanned aerial vehicle subject to wind disturbances and whose sensors parameters have been identified from flight tests data.

Performance Degradation Due to Particle Impoverishment in Particle Filtering

Particle filtering (PF) has shown its outperforming results compared to that of classical Kalman filtering (KF), particularly for highly nonlinear problems. However, PF may not be universally superior to the extended KF (EKF) although the case (i.e. an example that the EKF outperforms PF) is seldom reported in the literature. Particularly, PF approaches show degraded performance for problems where the state noise is very small or zero. This is because particles become identical within a few iterations, which is so called particle impoverishment (PI) phenomenon; consequently, no matter how many particles are employed, we do not have particle diversity regardless of if the impoverished particle is close to the true state value or not. In this paper, we investigate this PI phenomenon, and show an example problem where a classical KF approach outperforms PF approaches in terms of mean squared error (MSE) criterion. Furthermore, we compare the processing speed of the EKF and PF approaches, and show the better speed performance of classical EKF approaches. Therefore, PF approaches may not be always better option than the classical EKF for nonlinear problems. Specifically, we show the outperforming result of unscented Kalman filter compared to that of PF approaches (which are shown in Fig. 7(c) for processing speed performance, and Fig. 6 for MSE performance in the paper).

Descriptor-Type Kalman Filter and TLS EXIN Speed Estimate for Sensorless Control of a Linear Induction Motor

This paper proposes a speed observer for linear induction motors (LIMs), which is composed of two parts: 1) a linear Kalman filter (KF) for the online estimation of the inductor currents and induced part flux linkage components; and 2) a speed estimator based on the total least squares (TLS) EXIN neuron. The TLS estimator receives as inputs the state variables, estimated by the KF, and provides as output the LIM linear speed, which is fed back to the KF and the control system. The KF is based on the classic space-vector model of the rotating induction machine. The end effects of the LIMs have been considered an uncertainty treated by the KF. The TLS EXIN neuron has been used to compute, in recursive form, the machine linear speed online since it is the only neural network able to solve online, in a recursive form, a TLS problem. The proposed KF TLS speed estimator has been tested experimentally on a suitably developed test setup, and it has been compared with the classic extended KF.

A Method for Tracking Dim Target Using Particle Filter

Traditional methods encountered two serious problems in tracking dim targets. One is the nonlinearity of the system model, and other is the low SNR of measurement signals. The two problems are hardly solved simultaneously in practical engineering applications. The particle filter is a recursive numerical technique which uses random sampling to approximate the optimal evaluation to target tracking problems. In this paper, we developed a method for tracking dim target using particle filter. Simulation results showed that the tracking performance of this method has greatly improved compared with classical extended Kalman filter and unscented Kalman filter.

Experimental Study on Impact-Induced Damage Detection Using an Improved Extended Kalman Filter

This paper presents an experimental study on using an improved extended Kalman filter (EKF) to identify impact-induced structural damage. By introducing the optimization of estimated residual error into the classical EKF, this real-time approach demonstrates an excellent capability to identify the abrupt changes of structural parameters instantly and accurately. The optimization procedure is activated when a prescribed threshold is exceeded. A shaking table test of a three-story steel frame subjected to abrupt damage induced by impact load was conducted to validate the improved EKF approach. The results clearly reveal its improved performance and good anti-noise ability in identifying time-variant structural parameters.

Pipelined Particle Filter with Nonobservability Measure for Attitude and Velocity Estimation

In this paper, a particle filter is proposed to estimate attitude and velocity onboard a small unmanned aerial system using GPS, gyro, and accelerometer measurements. For the proposed filter setup, attitude observability heavily depends on the vehicle trajectory. Therefore, a measure of nonobservability is introduced to track the uncertainty in the attitude estimate. For the nonlinear estimation problem at hand, the use of a particle filter instead of a classical extended Kalman filter has shown to be advantageous in terms of robustness. However, particle filters are computationally very demanding, and thus, are usually not applicable for navigation onboard small unmanned aerial systems. To overcome this problem, an onboard computer system has been developed, which comprises a field programmable gate array as coprocessor. To get a quantitative measure on the performance of the proposed algorithm, the field-programmable-gate-array-based onboard computer system running a pipelined particle filter for attitude and velocity estimation has been flight tested on a civil-aviation aircraft equipped with a high-precision attitude and heading reference system. Further tests have been carried out with a fixed-wing small unmanned aerial system, in which the pipelined particle filter has been used as attitude reference in a simple waypoint-navigation scenario.

Real-Time Attitude-Independent Three-Axis Magnetometer Calibration for Spinning Projectiles: A Sliding Window Approach

This brief deals with attitude-independent three-axis magnetometer calibration in real time. The first part of this note presents a new model used for real-time estimation without any knowledge of the attitude of the projectile. The second part describes the other main contribution, which consists of a new simple and useful filter used for parameters estimation. The latter is formulated within the framework of an extended Kalman filter (EKF), which includes previous measurements in order to increase the performance and stability of the estimator. Finally, numerical simulations support the contribution of the presented algorithm. A good performance is shown through a smart projectile example in a noisy context, without requiring much more computation time than the classical EKF.

Sparsity-Promoting Extended Kalman Filtering for Target Tracking in Wireless Sensor Networks

In this letter, we study the problem of target tracking based on energy readings of sensors. We minimize the estimation error by using an extended Kalman filter (EKF). The Kalman gain matrix is obtained as the solution to an optimization problem in which a sparsity-promoting penalty function is added to the objective. The added term penalizes the number of nonzero columns of the Kalman gain matrix, which corresponds to the number of active sensors. By using a sparse Kalman gain matrix only a few sensors send their measurements to the fusion center, thereby saving energy. Simulation results show that an EKF with a sparse Kalman gain matrix can achieve tracking performance that is very close to that of the classical EKF, where all sensors transmit to the fusion center.

Some Relations Between Extended and Unscented Kalman Filters

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

Dynamic State Estimation in Power System by Applying the Extended Kalman Filter With Unknown Inputs to Phasor Measurements

Availability of the synchronous machine angle and speed variables give us an accurate picture of the overall condition of power networks leading therefore to an improved situational awareness by system operators. In addition, they would be essential in developing local and global control schemes aimed at enhancing system stability and reliability. In this paper, the extended Kalman filter (EKF) technique for dynamic state estimation of a synchronous machine using phasor measurement unit (PMU) quantities is developed. The simulation results of the EKF approach show the accuracy of the resulting state estimates. However, the traditional EKF method requires that all externally observed variables, including input signals, be measured or available, which may not always be the case. In synchronous machines, for example, the exciter output voltage Efd may not be available for measuring in all cases. As a result, the extended Kalman filter with unknown inputs, referred to as EKF-UI, is proposed for identifying and estimating the states and the unknown inputs of the synchronous machine simultaneously. Simulation results demonstrate the efficiency and accuracy of the EKF-UI method under noisy or fault conditions, compared to the classic EKF approach and confirms its great potential in cases where there is no access to the input signals of the system.

Unscented Kalman filter for power system dynamic state estimation

A new estimation method for power system dynamic state estimation, the unscented Kalman filter (UKF), is presented. It is based on the application of the unscented transformation (UT) combined with the Kalman filter theory. One of the challenges in the process of power system estimation is coping with a highly non-linear mathematical model of network equations, which is usually approximated through a linearisation. The new derivative free estimation method overcomes this limitation using the UT and achieves better accuracy with simpler implementation. The UKF is derived and demonstrated using three different test power systems under typical network and measurement conditions. Its performance is compared with the classical extended Kalman filter. The simplicity of the new estimator and its low computational demand make it a better option to be applied in the next generation of dynamic system estimators.

CI‐Graph simultaneous localization and mapping for three‐dimensional reconstruction of large and complex environments using a multicamera system

AbstractSubmapping and graphical methods have been shown to be valuable approaches to simultaneous localization and mapping (SLAM), providing significant advantages over the classical extended Kalman filter (EKF) solution: they are faster and, when using local coordinates, produce more consistent estimates. The main contribution of this paper is CI‐Graph SLAM, a novel algorithm that is able to efficiently map large environments by building a graph of submaps and a spanning tree of this graph with the following properties: (1) any pair of neighboring submaps in the spanning tree are conditionally independent and (2) the current submap is always up to date, containing the marginal probabilities of the submap variables given all previous measurements. Thanks to these properties, an old submap can be updated at any time by performing a single propagation from the current map to the old submap along the spanning tree. This operation is required only when a map is revisited, with a cost linear with the number of maps in the loop. At the end of the experiment the method performs a single propagation through the whole tree, recovering exactly the same marginals for all the map variables as the EKF–SLAM algorithm does, without ever needing to compute the whole covariance matrix. To evaluate CI‐Graph performance in extremely loopy environments, the method was tested using a synthetic Manhattan world. The behavior of the algorithm in large real environments is shown using the public data sets from the RAWSEEDS project in which a robot equipped with a trinocular camera traversed indoor and outdoor environments with several loops and revisited areas. Loops are robustly closed using a novel technique that detects candidate loop closures using a visual vocabulary tree and filters them using temporal and geometric constraints. Our experiments show that when using frontal cameras, the technique outperforms FAB‐MAP. The epipolar geometry of the loop‐closing images is used to find feature matches that are imposed on the CI‐Graph to correct the submap estimations along the loop. © 2010 Wiley Periodicals, Inc.

Experimental airborne NH∞ vision-based simultaneous localization and mapping in unknown environments

This article aims to present a robust airborne three-dimensional (3D) visual simultaneous localization and mapping (VSLAM) solution based on a stereovision system embedded in a remotely piloted helicopter. The proposed solution enables an unmanned aerial vehicle to construct a reliable map and localize itself in this map without any user intervention. The main contributions of this article are threefold: first, based on an adaptive scale-invariant feature transform algorithm, a robust approach for feature extraction and matching is proposed. Second, an adaptation of the non-linear H∞ filtering scheme is presented to the VSLAM problem, for which a 3D vision-based observation model is suggested. This filtering scheme avoids issues linked with the classical extended Kalman filtering techniques such as linearization errors, initialization problems, and noise statistics assumptions. Finally, a reliable map management approach, based on the k-nearest landmark concept, is presented to allow efficient loop closing detection and map building. This approach reduces significantly the complexity of the airborne VSLAM algorithm by making it independent of the landmark number. Experimental results using real data sets show the performance and the robustness of the proposed airborne VSLAM solution to build a reliable map and localize the unmanned aerial vehicle within this map.